How to Check If Two Numbers Are Co-Prime?
The concept of co-prime numbers plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding co-prime numbers helps students solve HCF/LCM word problems, recognize number properties, and avoid common mistakes in competitive exams. Let's explore the world of co-prime numbers in detail, with simple explanations, stepwise checks, solved examples, and Vedantu's easy shortcuts for mastering the topic.
What Is Co-Prime Numbers?
Co-prime numbers (also called relatively prime numbers) are any two natural numbers that have no common factor other than 1. That means, if the highest common factor (HCF) or greatest common divisor (GCD) of two numbers is 1, they are co-prime. For example, 8 and 15 are co-prime since their only common factor is 1. You’ll find this concept applied in areas such as HCF/LCM calculations, rational numbers, and divisibility problems.
Key Formula for Co-Prime Numbers
Here’s the standard formula: If GCD(a, b) = 1, then a and b are co-prime numbers.
Co-Prime Numbers Examples
| Pair of Numbers | Are They Co-Prime? | Why? |
|---|---|---|
| 5, 7 | Yes | No common factor except 1 |
| 8, 15 | Yes | No common factor except 1 |
| 9, 12 | No | Common factor: 3 |
| 14, 15 | Yes | No common factor except 1 |
| 12, 18 | No | Common factor: 6 |
| 17, 19 | Yes | No common factor except 1 (both prime) |
How to Check if Two Numbers are Co-Prime?
Follow this easy step-by-step method to check co-primality:
- List all factors of both numbers.
- Check for any common factor other than 1.
- If there is no other common factor, the pair is co-prime.
- You can also use the HCF/GCD method — if HCF(a, b) = 1, they are co-prime.
Example: Are 18 and 25 co-prime?
1. Factors of 18: 1, 2, 3, 6, 9, 182. Factors of 25: 1, 5, 25
3. Common factors: only 1
4. Since no other common factor exists, 18 and 25 are co-prime.
Co-Prime Numbers from 1 to 100
Here are some popular co-prime pairs in the range 1 to 100, useful for quick school revision and worksheets:
| Co-Prime Pair | Reason |
|---|---|
| (1, 99) | 1 is co-prime with every number |
| (14, 15) | Consecutive numbers are always co-prime |
| (17, 60) | No common factor except 1 |
| (12, 25) | No common factor except 1 |
| (99, 100) | Consecutive numbers are always co-prime |
| (29, 31) | Both are prime, so automatically co-prime |
| (18, 35) | No common factor except 1 |
Prime Numbers vs. Co-Prime Numbers
| Prime Numbers | Co-Prime Numbers |
|---|---|
| A number that has only two factors: 1 and itself | A pair of numbers that have no common factor except 1 |
| E.g., 2, 3, 5, 7, 11 | E.g., (4, 9), (8, 15), (21, 22) |
| Prime is a property of a single number | Co-prime is a property of a pair (or group) of numbers |
| Every pair of primes is co-prime | But co-prime numbers need not be prime |
Step-by-Step Illustration
Let’s check if 16 and 27 are co-prime:
1. List factors of 16: 1, 2, 4, 8, 162. List factors of 27: 1, 3, 9, 27
3. Common factor: Only 1
4. Result: Since HCF(16, 27) = 1, they are co-prime numbers
Speed Trick or Vedic Shortcut
Here’s a quick shortcut: Two numbers are always co-prime if they are consecutive (like 35, 36), or if one is an odd number and the other is an even number not divisible by the same base factors. Use the HCF trick: Try dividing both numbers by 2, 3, 5, etc. If nothing matches except 1, they are co-prime.
Example Trick: To check if 51 and 80 are co-prime, check divisibility by 2, 3, 5, 7 (small primes). None match. Their HCF = 1. Answer: Co-prime!
Vedantu’s live classes often showcase more number hacks for school and Olympiad problems.
Try These Yourself
- Write five pairs of co-prime numbers between 1 and 50.
- Check if (44, 99) is a co-prime pair.
- Find all co-prime pairs from 28 to 34.
- Spot which among (18, 49), (21, 28), (40, 41) is not a co-prime pair.
Frequent Errors and Misunderstandings
- Confusing co-prime numbers with prime numbers (not all co-prime numbers are primes).
- Assuming two even numbers can be co-prime — except for (2, 1), two even numbers are never co-prime.
- Forgetting that 1 is co-prime with every number.
Relation to Other Maths Concepts
The idea of co-prime numbers connects closely with concepts like Highest Common Factor (HCF) and Lowest common multiple (LCM). Mastering this helps with Factors and Multiples and Prime Factorization—all of which are essential for JEE, NTSE, and school exams.
Classroom Tip
A great way to remember co-prime numbers is: If two numbers have "1" as their only common factor—they’re co-prime! Vedantu’s teachers use simple table hacks and divisibility games to build your co-prime skills in fun live sessions.
We explored co-prime numbers—from definition, formula, tables, mistakes, to connections with related topics. Keep practicing with Vedantu and grow confident in spotting and using co-prime numbers in Maths problems and real life.
Explore related topics: Prime Numbers | Factors and Multiples| Prime Factorization
FAQs on Co-Prime Numbers Explained Simply with Examples
1. What are co-prime numbers with examples?
Co-prime numbers (also called relatively prime numbers) are pairs of numbers whose greatest common divisor (GCD) or highest common factor (HCF) is 1. This means they do not share any factor apart from 1.
- Example: The numbers 4 and 9 are co-prime because the only factor they share is 1.
- Another example: 8 and 15 are co-prime since their HCF is 1.
2. What are the co-prime numbers from 1 to 100?
A pair of numbers between 1 and 100 is said to be co-prime if their only common factor is 1. Some examples of co-prime pairs between 1 and 100 are:
- 14 and 25
- 35 and 48
- 17 and 68
3. Is 18 and 25 a Coprime?
Yes, 18 and 25 are co-prime numbers.
- Prime factorization of 18: 2 × 3 × 3
- Prime factorization of 25: 5 × 5
4. Are 15 and 37 co-prime?
Yes, 15 and 37 are co-prime numbers. Let's check:
- Prime factors of 15: 3 × 5
- Prime factors of 37: 37 (since 37 is a prime number)
5. What is the difference between co-prime numbers and prime numbers?
Co-prime numbers are pairs of numbers whose only common factor is 1, regardless of whether each number is prime.
Prime numbers, on the other hand, are numbers greater than 1 that have no divisors other than 1 and themselves.
- Example of co-prime but not prime: 8 and 15 are both composite, but they are co-prime.
- Example of two primes: 5 and 7 are both prime, so they are always co-prime.
6. How do you find if two numbers are co-prime using the HCF method?
To check if two numbers are co-prime, calculate their Highest Common Factor (HCF):
- Find the HCF (or GCD) using the Euclidean algorithm or prime factorization.
- If the HCF is 1, the numbers are co-prime.
- Prime factors of 14: 2 × 7
- Prime factors of 25: 5 × 5
7. Can consecutive numbers be co-prime? Why?
Yes, consecutive numbers (any two numbers that follow each other, like 11 and 12) are always co-prime. This is because consecutive numbers never share a common factor other than 1.
- For example, HCF(14, 15) = 1.
8. Why are co-prime numbers important in mathematics?
Co-prime numbers play a crucial role in various branches of mathematics, including:
- Fractions: Reducing fractions to lowest terms requires numerator and denominator to be co-prime.
- Encryption: Co-prime numbers are vital in cryptography and coding theory.
- Number theory: They are the foundation for Euler’s Totient Function and modular arithmetic.
9. Are two even numbers ever co-prime?
No, two even numbers can never be co-prime because both are divisible by 2. Their HCF will always be at least 2, not 1.
- For example, 4 and 6 share a factor of 2.
